A New Dual System For The Fundamental Units, including and going beyond the newly revised SI
aa r X i v : . [ phy s i c s . g e n - ph ] J u l A NEW DUAL SYSTEM FOR THE FUNDAMENTAL UNITS including and going beyond the newly revised SI P ierre Fayet
Laboratoire de physique de l’ ´Ecole normale sup´erieure24 rue Lhomond, 75231 Paris cedex 05, France ∗ and Centre de physique th´eorique, ´Ecole polytechnique, 91128 Palaiseau cedex, France Abstract
We propose a new system for the fundamental units, which includes and goes beyond the presentredefinition of the SI, by choosing also c = ~ = 1. By fixing c = c ◦ m/s = 1, ~ = ~ ◦ J s = 1and µ ◦ = µ ◦ N/A = 1, it allows us to define the metre, the joule, and the ampere as equal to(1/299 792 458) s, (1 / ~ ◦ = . ... × ) s − and √ µ ◦ N = p µ ◦ c ◦ / ~ ◦ s − = 1 . ... × s − .It presents at the same time the advantages and elegance of a system with ~ = c = µ ◦ = ǫ ◦ = k = N A = 1 , where the vacuum magnetic permeability, electric permittivity, and impedance are allequal to 1.All units are rescaled from the natural ones and proportional to the s, s − , s − , ... or just 1, as forthe coulomb, ohm and weber, now dimensionless. The coulomb is equal to p µ ◦ c ◦ / ~ ◦ = 1 . ... × ,and the elementary charge to e = 1 . ... × − C = √ πα = . ... . The ohm is equal to 1 /µ ◦ c ◦ so that the impedance of the vacuum is Z ◦ = 376 . ... Ω = 1 . The volt is 1 / √ µ ◦ c ◦ ~ ◦ s − =5 . ... × s − , and the tesla c ◦ V/m = p c ◦ /µ ◦ ~ ◦ s − = 4 . ... × s − .The weber is 1 / √ µ ◦ c ◦ ~ ◦ = 5 . ... × . The flux quantum is Φ ◦ = h/ e = 2 . ... × − Wb= π/e = 10 . ... , with K J = 483 597 . ... GHz/V = e/π = .09639 ..., and R K = 25 812 . ... Ω = 1 / α ≃ .
518 . One can also fix e = 1 .
602 176 634 × − C, at the price of adjusting the coulomb and allelectrical units with µ ◦ = 4 π × − η where η , ∝ α , is very close to 1. R´esum´e
Nous proposons un nouveau syst`eme pour les unit´es fondamentales, qui inclut la red´efinition encours du SI et va au-del`a, en choisissant aussi c = ~ = 1. En fixant c = c ◦ m/s = 1, ~ = ~ ◦ J s= 1 et µ ◦ = µ ◦ N/A = 1, il permet de d´efinir le m`etre, le joule et l’amp`ere comme ´egaux `a(1/299 792 458) s, (1 / ~ ◦ = 0 , ... × ) s − et √ µ ◦ N = p µ ◦ c ◦ / ~ ◦ s − = 1 , ... × s − . Ilpr´esente en mˆeme temps les avantages et l’´el´egance d’un syst`eme o`u ~ = c = µ ◦ = ǫ ◦ = k = N A = 1 ,o`u la perm´eabilit´e magn´etique, la permittivit´e ´electrique et l’imp´edance du vide sont ´egales `a 1.Toutes les unit´es sont red´efinies `a partir des unit´es naturelles et proportionnelles `a la seconde, s − ,s − , ... ou `a 1, comme pour le coulomb, l’ohm et le weber, sans dimensions. Le coulomb est ´egal `a p µ ◦ c ◦ / ~ ◦ = 1 , ... × , et la charge ´el´ementaire `a e = 1 , ... × − C = √ πα = 0 , ... .L’ohm est ´egal `a 1 /µ ◦ c ◦ , et l’imp´edance du vide `a Z ◦ = 376 , ... Ω = 1. Le volt est 1 / √ µ ◦ c ◦ ~ ◦ s − = 5 , ... × s − , et le tesla c ◦ V/m = p c ◦ /µ ◦ ~ ◦ s − = 4 , ... × s − .Le weber est 1 / √ µ ◦ c ◦ ~ ◦ = 5 , ... × , et le quantum de flux Φ ◦ = h/ e = 2 , ... × − Wb= π/e = 10 , ... , avec K J = 483 597 . ... GHz/V = e/π = 0,09639 ... , et R K = 25 812 . ... Ω =1 / α ≃ ,
518 . On peut aussi fixer e = 1 ,
602 176 634 × − C `a condition d’ajuster le coulombet toutes les unit´es ´electriques avec µ ◦ = 4 π × − η o`u η , ∝ α , est tr`es proche de 1.LPTENS/18/19 ∗ . LPENS, ´Ecole normale sup´erieure, Universit´e PSL ; CNRS UMR 8023 ; Sorbonne Universit´e ; UPD-USPC
I. PRESENTATION OF THE NEW SYSTEM
Measuring geometrical or physical quantities, such as intervals of time, space distances, masses, energies,electric charges and currents, etc., requires the definition of appropriate units. It is desirable that theseunits be the same everywhere, and do not change with time. To realize this they must be defined asmuch as possible in a universal way, from invariant physical objects or phenomena. This is how the metrewas initially defined as “la dix-millioni`eme partie du quart du m´eridien terrestre” [1]. The unit of weightfollowed, as the weight of a dm of water at the temperature of melting ice [2]. This was at the origin ofthe metric system in France at the end of the XVIIIth century, and later of the “Convention du m`etre”in 1875 [3], leading to the International System of Units (SI) [4, 5].As of today, the second is defined from the period of a specific transition of the caesium-133 atom [6].The metre is derived from the second by fixing the value of the speed of light in vacuum, c , to be exactly299 792 458 m/s [7]. This is made possible thanks to the theory of relativity, verified to a very high degreeof precision, according to which time and space are related entities of similar nature, the speed of lightbeing the same, always and everywhere, independently of the reference frame in which it is measured.But the unit of mass, the kilogram, still remains defined from a physical artefact, as the mass of theinternational prototype of the kilogram ( K ), a cylinder of platinum-iridium alloy stored at the Inter-national Bureau of Weights and Measures (BIPM) in the “Pavillon de Breteuil” at S`evres, France [8].Its mass remains constant and equal to 1 kg by definition and according to an international agreement,even if the corresponding quantity of matter cannot remain exactly constant due to surface effects anddoes inevitably vary very slightly over the years [9]. This is not a satisfactory situation, and it would bedesirable to get a universal definition of the kilogram, based on reproducible universal phenomena ratherthan on a single material object.This is the main purpose of the present redefinition of the kilogram, that will soon be derived fromthe second, or the metre, using quantum mechanics, by fixing the value of the Planck constant h [10].This one relates, through the relation E = hν (= ~ ω ), the energy E of a photon to the frequency ν (or angular frequency ω = 2 πν ) of the corresponding electromagnetic wave. Fixing h to a certainnumber of joule · seconds, in agreement with its presently measured value ( h ≃ . ... × − J s , or ~ = h/ π ≃ . ... × − J s) will allow us to define the unit of energy, the joule, as correspondingto a very large number of s − ( . ... × s − ). The definition of the kilogram follows, such that 1joule = 1 kg m s − as usual, rendering obsolete the international prototype stored at BIPM. Of coursethe value of h gets fixed in agreement with its present best determined value (6 .
626 070 15 × − J s[11]), so that the mass of the international prototype will still be practically equal to 1 kg at the time ofthe change, within experimental uncertainties. The change will not affect in any practical way the valuesof masses at the time it is performed, while allowing for more precise measurements in the future.We could choose to measure distances directly in seconds by fixing the speed of light at c = 1, thenatural choice in relativity ; and energies and masses directly in s − by fixing the value of ~ = h/ π also at 1, the natural choice in quantum mechanics. This would be conceptually much simpler. Butconsidering the second as being also a unit of distance, close to 3 × km, and the second − as acommon unit for energy and mass (with c = 1), would not be very practical. Indeed the energy unitof 1 s − would be very small, 1 . ... × − joule. Furthermore, the kilogram being itself associatedwith a very large energy E = mc = . ... × J, the s − as a unit of mass would be comparativelyeven smaller, (1 . ... × − ) / ( . ... × ) kg i.e. 1 . ... × − kg. This is also understood from ~ /c ≃ . ... × − kg which fixes the mass unit of 1 s − at 1 . ... × − kg.While such units are often very convenient in relativistic and quantum physics, it would seem unprac-tical to replace square metres by ≈ − s when measuring surfaces, or to ask for 10 s − of potatoeson the market. For historical and practical reasons we remain attached to measuring distances in metresrather than in seconds, and masses and energies in kilograms and joules rather than in s − . Fortunatelywe still have the freedom to define the metre, kilogram, and joule as corresponding to fixed submultiplesor multiples of the second, or second − . This is how the units of length, and very soon energy and mass,are or will be derived from the unit of time by fixing c and ~ to a certain number of m/s and J s accordingto c = c ◦ m/s, ~ = ~ ◦ J s, allowing to define the metre and the joule, and subsequently the kilogram, thenewton, ... , from the second. This is what the present redefinition of the SI, to become effective from 20May 2019, is going to achieve [10].But we can still go further. Within relativity space and time, related by the Lorentz symmetry group,become quantities of similar nature, which may be measured with a common unit. It is the same for massand energy. Within quantum mechanics energies and angular frequencies are directly related, allowing usto measure energies in s − , as for angular frequencies. In a conceptually ideal system we ought to choose c = 1 as suggested by relativity, and ~ = 1 as suggested by quantum mechanics so that the intrinsicangular momentum, or spin, of the electron is simply 1/2. This is in fact implicitly supposed when wesay that the electron is a spin-1/2 particle. Such a system, however, is usually viewed as an idealizedone for theoreticians, requiring to move back to an ordinary system in which time, space, energies andmasses are all measured in terms of their own units, in seconds, metres, joules and kilograms, as in thetraditional or newly revised SI.We shall show here that the two points of view, although apparently antagonist, may be reconci-led within a single unified system including the new SI , in which both c and ~ are also equal to 1 .Then the metre, joule and kilogram get identified as specific submultiples or multiples of the second orsecond − , the natural units of length, energy, and mass in an ideal system with c = ~ = 1, once thesecond is chosen as the unit of time. We can thus have, simultaneously, ( c = 299 792 458 m/s and c = 1 ,h = 6 .
626 070 15 × − J s and ~ = h/ π = 1 . (1)These equalities will provide as desired the appropriate new SI definitions for the metre, the joule, andthe kilogram, also made compatible with the advantages of working in a system with c = ~ = 1, in which the fundamental laws of physics do not even depend on these parameters . These were previously referredto as “fundamental constants of nature”. Once identified with unity, they get downgraded to simplyproviding numerical conversion factors between related units. II. THE NEW SYSTEM FOR THE ELECTRICAL UNITS
This new picture can be extended to the electrical units, up to now defined from the mechanical onesby fixing the value of the parameter µ ◦ defining the magnetic permeability of vacuum and entering thetraditional definition of the ampere [4, 12, 13]. The new approach we propose here remains applicable,as we shall see, in the context of the new SI [10], in which the rigid connection between electrical andmechanical units gets somewhat loosened. This is a consequence of the recent decision taken at the lastCGPM to define the coulomb (and thus the ampere and all electrical units) so that the numerical valueof the elementary charge is exactly fixed, at e = 1 .
602 176 634 × − C, requiring to turn µ ◦ into anadjustable parameter [14]. It is indeed the price to pay for having decided to fix the numerical value of e , rather than measuring it as it has been done up to now.Let us return to our new proposed approach. The present conventional choice µ ◦ = 4 π × − N/A ,or equivalently H/m, before the upcoming 2019 redefinition of electrical units, may be reconciled withthe ideal one µ ◦ = 1 by fixing the value of µ ◦ , as done for c and ~ , at a certain number of N/A . Weshall thus write µ ◦ = µ ◦ N/A , where µ ◦ , now dimensionless, is initially fixed to 4 π × − . But, as from2019 the coulomb will no longer be obtained from the traditional definition of the ampere [4, 12, 13] butby fixing the value of the elementary charge [10], the value of µ ◦ will still have be very slightly adjusted,from 4 π × − into µ ◦ = 4 π × − η . In both cases , and thus independently of the on-going 2018-2019 change for the definition of theelectrical units, we can ask for µ ◦ = 1, which provides an ideal system with ~ = c = µ ◦ = ǫ ◦ = 1. Fixing µ ◦ = µ ◦ N / A = 1 determines the ampere, which must verify1 A = µ ◦ N . (2)The ampere appears as proportional to a square root of the newton , which is conveniently expressed as1 A = √ µ ◦ N .Altogether the set of three equations c = c ◦ m / s = 1 , which fixes the metre, ~ = ~ ◦ J s = 1 , which fixes the joule, µ ◦ = µ ◦ N / A = 1 , which fixes the ampere (3)leads to a system that includes and goes beyond the present redefinition of the SI. It allows us to reconcile,within a single system of units, the advantages of a conceptually desirable system in which ~ = c = µ ◦ = ǫ ◦ = 1, the vacuummagnetic permeability, electric permittivity, and impedance being all equal to 1, with the elementarycharge e = √ πα = . ... , K J = 2 e/h = e/π and R K = h/e = (376 . ... Ω) / α = 1 / α ; the convenience of our familiar SI units, second, metre, kilogram, joule, newton, ampere, coulomb,volt, etc. appropriately defined or redefined very much as usual ; with all these units also expressed in terms of the second, s − , s − , ... , or even as pure numbersas for the ohm, the coulomb, and the weber ; furthermore, as the numerical value of e will also get fixed, the coulomb, the ampere and all electricalunits get redefined accordingly, using an adapted value of µ ◦ equal to 4 π × − η where η is very closeto 1 ; this adjustment of electrical units is automatically and explicitly taken care of by expressing theampere and coulomb proportionally to √ µ ◦ , as indicated above.With the values of ~ and c getting fixed to ~ ◦ J s and c ◦ m/s, respectively, we get the expression of thenewton 1 N = 1 J s1 m / s s − = c ◦ ~ ◦ s − . (4)This leads, most notably, to the following dual expressions for the ampere A and the coulomb C (at themoment still evaluated with µ ◦ = 4 π × − ), the elementary charge e and the impedance of the vacuum Z ◦ : p µ ◦ N = p µ ◦ c ◦ / ~ ◦ s − = 1 .
890 067 014 853 ... × s − , p µ ◦ kg m = 1 / p ǫ ◦ c ◦ ~ ◦ = 1 .
890 067 014 853 ... × ,e = 1 .
602 176 634 × − C = √ πα ≃ .
302 822 1208 ,Z ◦ = µ ◦ c ◦ Ω = 376 .
730 313 461 ...
Ω = 1 . (5)The elementary charge e remains as an independent dimensionless free parameter equal to √ πα , to bedetermined experimentally . However, it will get numerically fixed at exactly 1 .
602 176 634 × − C inthe new SI, at the price of suitably adjusting the coulomb and thus the value of µ ◦ , now to be taken as4 π × − η .The electron-volt, 1 eV = 1.602 ... × − J , gets exactly fixed at ( e ◦ / ~ ◦ ) s − = 1 . ... × s − .The volt, equal to 1 / √ µ ◦ c ◦ ~ ◦ s − , is also recovered as 1 eV/ e = 1 . ... × s − /. ... = 5 . ... × s − . The Josephson and von Klitzing constants are related, respectively, to the size of the electron-volt and to the fine structure constant α . They get fixed numerically in SI units (GHz/V and ohms) [14]thanks to the choice of h = h ◦ J s, e = e ◦ C. At the same time they also become pure numbers equal to e/π and 1 / α , respectively, thanks to our choice of ~ = 1 with c = µ ◦ = 1 so that e = √ πα : K J = 2 e/h = 2 e ◦ /h ◦ Wb − = 483 597 .
848 416 ...
GHz / V = e/π ≃ . ,R K = h/e = h ◦ e ◦ Ω = µ ◦ c ◦ Ω2 α = 25 812 .
807 459 ...
Ω = 1 / α ≃ .
517 999 58 . (6) K J = e/π is equivalent to saying that 1 eV = π (2 e ◦ /h ◦ ) s − = π × .
483 597 ... × s − = 1 . ... × s − . The SI value in ohms of R K is the vacuum impedance, Z ◦ = µ ◦ c ◦ Ω = 376 .
730 313 ...
Ω (nowalso equal to 1), multiplied by 137 .
035 999 ... / . ... Ω . It is now, atthe same time, also dimensionless and equal to 137. 035 999 ... /2.
Table
I: The new dual system : c = c ◦ m / s = 1 and ~ = ~ ◦ J s = 1 fix the metre and the joule. µ ◦ = µ ◦ N / A = 1allows us to express the ampere and coulomb as 1 A = √ µ ◦ N and 1 C = √ µ ◦ kg m . With ~ /c = ( ~ ◦ /c ◦ ) kg m = 1,the coulomb, dimensionless, is equal to p µ ◦ c ◦ / ~ ◦ = 1 . ... × . The elementary charge is e = 1 . ... × − C= √ πα = . ... , and the impedance of vacuum Z ◦ = µ ◦ c ◦ Ω = 376 . ... Ω = 1, with K J = 2 e/h = e/π and R K = h/e = 1 / α . The dimensionless coulomb, ohm and weber are related by 1 C × √ µ ◦ N. The two other columns givetheir expressions in s, s − , s − or as dimensionless constants. This is the case for the coulomb, ohm, and weber,and for e = √ πα , the impedance of vacuum Z ◦ = 1, and flux quantum Φ ◦ = π/e . The numerical values areevaluated here for µ ◦ = 4 π × − , and the corresponding benchmark value α = α ◦ = 1 /
137 035 999 158 713 ... asin (14). They should be rescaled using the parameter η = p α/α ◦ according to (18), and may change by a few10 − , depending on α . Expressions in terms of s, s − , s − or as constants“Conventional expressions” ( ~ = c = µ ◦ = ǫ ◦ = 1)1 m = 1 c ◦ s = 1299 792 458 s1 J = 1 ~ ◦ s − = .
948 252 156 246 ... × s − c ◦ ~ ◦ s − = .
852 246 536 175 ... × s − c ◦ ~ ◦ s − = 2 .
842 788 447 250 ... × s − √ µ ◦ N = r µ ◦ c ◦ ~ ◦ s − = 1 .
890 067 014 853 ... × s − √ µ ◦ kg m = r µ ◦ c ◦ ~ ◦ = 1 .
890 067 014 853 ... × / C = 1 √ µ ◦ c ◦ ~ ◦ s − = 5 .
017 029 284 119 ... × s − / C = 1 √ µ ◦ q J/m m / s = r c ◦ µ ◦ ~ ◦ s − = 1 .
504 067 540 944 ... × s − / (A m) = 1 √ µ ◦ q J/m = s c ◦ µ ◦ ~ ◦ s − = 4 .
509 081 050 976 ... × s − √ µ ◦ c ◦ ~ ◦ = 5 .
017 029 284 119 ... × ............................................................................................................................................................................... / V = 1 s / Ω = µ ◦ s / m = µ ◦ c ◦ s = 376 .
730 313 461 ... s1 H = 1 J/A = 1 Ω s = 1 µ ◦ m = 1 µ ◦ c ◦ s = 1 / .
730 313 461 ... s1 Ω = 1 V / A = 1 W / A = 1 µ ◦ m / s = 1 µ ◦ c ◦ = 1 / .
730 313 461 ...Z ◦ = µ ◦ c ◦ Ω = 376 .
730 313 461 ...
Ω = µ ◦ c = 1 e = 1 .
602 176 634 × − C = √ πα = .
302 822 120 789 ... .
602 176 634 × − J = e ◦ ~ ◦ s − = 1 .
519 267 447 878 ... × s − Φ ◦ = h/ e = 2 .
067 833 848 461 ... × − Wb = r π α = πe = 10 .
374 382 972 ...K J = 2 e/h = 483 597 .
848 416 983 ...
GHz / V = r απ = eπ = . ...R K = he = µ ◦ c ◦ Ω2 α = 25 812 .
807 459 304 ...
Ω = µ ◦ c α = 12 α = 68 .
517 999 579 ...
The resulting expressions of the various mechanical and electrical units are given in Table I. We leaveaside for the time being the question of a more fundamental definition for the unit of time, that onemight like to obtain in the future from the electron mass m e using ~ /m e c ≃ .
288 088 667 × − s. III. THE IMPLICATIONS OF FIXING THE VALUE OF THE ELEMENTARY CHARGE
Let us discuss further the consequences of fixing exactly, in the planned revision of the SI to becomeeffective from 20 May 2019 [10], the numerical value of the elementary charge e when expressed incoulombs, according to the additional requirement e = e ◦ C = 1 .
602 176 634 × − C . (7)Although this may be conceptually questionable, this is intended to provide a very precise way to fix thecoulomb, and the other electrical units, allowing for more precise measurements.However, adopting the new expression (7) of the elementary charge e requires adjusting suitably the sizeof the coulomb, at present defined as 1 A s, and thus the size of the ampere. Writing the two expressionsof e according to the old and new definitions of the coulomb, e = e old C old = e ◦ η π × − C old = e ◦ C,we see that the redefined coulomb C may be “larger” (or possibly smaller) than the earlier one by a factor η = CC old = e old e ◦ , (8)very close to 1 . The coulomb, and thus the ampere, equal to √ µ ◦ N, get multiplied by η , with1 A = p µ ◦ N = p π × − N × η , (9)so that µ ◦ should be multiplied by η , becoming µ ◦ = 4 π × − η .We must ensure that the two concurrent definitions of the coulomb, and the ampere, are compatible.The ampere is expressed as 1 A = √ µ ◦ N as in (5). The compatibility of the set of 4 equations in (3,7)requires that the third equation in (3), µ ◦ = µ ◦ N / A = 1, be used, no longer to fix the exact size of theampere from a fixed µ ◦ = 4 π × − , but instead to determine a new “floating” value for µ ◦ , from nowon equal to 4 π × − η , from an ampere already defined as 1 C/s. The two definitions for the coulomb(and ampere) from (5) and (7) should be equalized, as follows1 A = p µ ◦ N = p µ ◦ c ◦ / ~ ◦ s − = ⇒ / p ǫ ◦ c ◦ ~ ◦ = e/e ◦ ⇐ = e = e ◦ C . (10)This provides two equivalent expressions for the fine structure constant, α = e ◦ πǫ ◦ ~ ◦ c ◦ = µ ◦ c ◦ e ◦ π ~ ◦ = e π . (11) α = e / π corresponds to the choice ~ = c = µ ◦ = ǫ ◦ = 1 also made here, leading to the dual expressionof the elementary charge e = e ◦ C = 1 .
602 176 634 × − C = √ πα ≃ .
302 822 1208 . (12)The price to pay for fixing e ◦ as above is that µ ◦ should now be multiplied by η , getting adjusted to µ ◦ = 4 π ~ ◦ c ◦ e ◦ α = 4 π × − η , (13)so that α stays unchanged. This leads to evaluate the benchmark value for α corresponding to keepingan unchanged µ ◦ = 4 π × − with the new fixed choice of e ◦ , α ◦ = 4 π × − c ◦ e ◦ π ~ ◦ = 7 .
297 352 565 305 ... × − = 1 / .
035 999 158 713 ... . (14)It is very close to the present best-determined value α = 1 / .
035 999 139 (31), with a relative standarduncertainty of 2 . × − [11, 15]. It corresponds to √ πα ◦ = .
302 822 120 789 201 ... . (15) µ ◦ is no longer rigidly fixed, but proportional to α . Fixing e ◦ as in (7,12) leads to a very small rescalingof all electrical units depending on the parameter η = p α/α ◦ ≃ < ∼ a few 10 − ) . (16)Thus fixing e = e ◦ C = ⇒ all electrical units become dependent on α , (17)with(A , C) ∝ η , (V , T , Wb) ∝ η − , F ∝ η , (H , Ω) ∝ η − ; µ ◦ ∝ η , ǫ ◦ ∝ η − ; with η = p α/α ◦ . (18)The sizes of the electrical units themselves become dependent on future experimental measurements ofthe fine structure constant α .In particular the value in ohms of the “impedance of the vacuum”, µ ◦ c ◦ ∝ α , now depends on α in thenew SI, i.e. on what was previously the value of the elementary charge e ! Of course Z ◦ itself, equal to µ ◦ c ◦ Ω, does not, as one can verify from (18). Still this artificially introduced dependence of the measureof the impedance of vacuum may look strange, especially for a quantity that is basically 1, with anappropriate choice of fundamental units (see below, subsection X B). Its measure µ ◦ c ◦ should actually beviewed as a parameter characterizing the size of the ohm (unfortunately dependent on the experimentallymeasured value of α in the new SI), and not as a measure of a physical property of the vacuum.This conceptual inconvenience of the new SI associated with the floating character of the electricalunits gets alleviated within the unified framework proposed here, in which the impedance of vacuum, Z ◦ = µ ◦ c ◦ Ω, while still equal to 376.730 ... Ω, is also identical to 1. The magnetic permeability ofvacuum, now expressed as µ ◦ = µ ◦ N / A = 1, remains also equal to 1 with 1 A = √ µ ◦ N, even if theampere A, as well as the numerical value µ ◦ of the vacuum permeability (expressed in N/A or H/m),are both dependent on α , as seen in (18).In case α were to vary with time as the possibility is occasionally considered, all electrical units,including the dimensionless ohm, coulomb, and weber, would also vary with time. This may be expressedthrough the set of equations˙AA = ˙CC = − ˙VV = − ˙TT = − ˙WbWb = 12 ˙ αα , ˙FF = − ˙HH = − ˙ΩΩ = ˙ αα , (19)as seen from eqs. (18) and Table I. This is a not-so-attractive consequence of the new SI choice of fixing e ◦ as in (7), in principle not compatible with the requirement that units should not change with time.Fortunately there are very strong limits on a possible time variation of α (with | ˙ α/α | constrained to beat present < − / y), so that in practice this does not appear as a limitation.Let us now return for some time to the mechanical units, defining the metre, the joule, and the kilogramby fixing the numerical values of c and ~ . IV. DEFINING THE METRE FROM THE SECOND, BY FIXING c The unit of time, the second, has long been defined as the fraction 1/86 400 of the “mean solar day”.It was then redefined from a physical phenomenon involving the period of a specific atomic transition ofthe caesium-133 atom. More precisely, since 1967 [6], “ The second is the duration of 9 192 631 770 periods of the radiation corresponding to the transitionbetween the two hyperfine levels of the ground state of the caesium-133 atom.” (20)The unit of length, the metre, first introduced as “la dix-millioni`eme partie du quart du m´eridienterrestre” [1], was realized as the “M`etre des Archives” in 1799, then defined from 1889 to 1960 as thelength of a specific object, the international prototype of platinum-iridium stored in the “Pavillon deBreteuil” at S`evres. It was replaced in 1960 by a definition involving the wavelength of a krypton-86radiation [16]. The theory of relativity, based on the invariance of the speed of light in vacuum, c , allowedfor a new definition of the metre by relating it to the unit of time. This was done in 1983 by fixing thevalue of c at, exactly, c = 299 792 458 m/s [7]. Since this date “ The metre is the length of the path travelled by light in vacuumduring a time interval of 1/299 792 458 of a second.” (21)This is now reformulated, in an equivalent way, by stating that [10] “ the speed of light in vacuum c is 299 792 458 m/s ” . (22)Indeed, relativity relates intimately the concepts of time and space, describing events in a 4-dimensionalspacetime with coordinates x µ = ( ct, ~x ). It also relates the energy E and momentum ~p of a particle ofmass m into the components of a 4-vector p µ = ( E/c, ~p ), with p µ p µ = E /c − ~p = m c . This reducesto the famous E = mc for a massive particle at rest, and E = pc for a massless photon travelling at thespeed of light. The unit of length may thus be derived from the unit of time by fixing the value of c . V. FIXING c = 299 792 458 m/s , OR c = 1 , OR BOTH AT THE SAME TIME Choosing c = 1 would be the most natural choice, leading to measure space distances directly inseconds, and energies and momenta in units of mass. The unit of length u l would then be the distancetravelled by light during the unit of time u t . Once this one is defined as the second, u l is the lengthtravelled by light in vacuum during 1 s (sometimes called a light-second), namely in SI units unit of length u l = c s = 299 792 458 m . (23)This is almost the distance to the Moon. Choosing c = 1 could lead to abandon the metre to replace itby a unit of length almost 300 million times larger, which is usually rejected as unpractical.In spite of that, we remain free to continue using the metre, now defined as a submultiple of the abovenatural unit of length according to 1 m = 1/299 792 458 u l . This expresses that the speed of light ischosen to be c = 1 u l / u t = 299 792 458 m/s.Even better, we can do both things at the same time by requiring c = 299 792 458 m/s and, “en mˆemetemps”, c = 1 . We can then complete the official formulation (22) as follows“ the speed of light in vacuum is c = 299 792 458 m/s ≡ . (24)It expresses that the natural unit of length according to relativity, u l = 299 792 458 m, is not independentfrom the unit of time, and may be identified with it. We then get1 m = 1299 792 458 s . (25)Expressing that c = c ◦ m/s ≡ /c ◦ ) s.This unconventional formulation defines directly the metre as a submultiple of the second. It may betaken as an improved definition of the metre, reconciling the present one in (21,22) with the choice c = 1suggested by relativity. Conversely, the metre also appears as a unit of time, as is natural owing to thesymmetry between space and time provided by relativity. We may then say that “ The time interval taken by light to travel in vacuum along a path of one metre longis 1/299 792 458 of a second, or ... one metre.” (26)We may even define a common subunit for distance and time appearing as a (metric) foot, equal to 1 ns= .299 792 458 m ≃ c = 1.We still need a unit of mass. This one is defined, since the first “Conf´erence G´en´erale des Poids etMesures” in 1889, as the mass of the international prototype of the kilogram (IPK), also known as “legrand K”, K , conserved at the BIPM. This was formulated in 1901 as [8] : “The kilogram is the unit of mass ; it is equal to the mass of the international prototype of the kilogram.” (27)The unit of force is then the newton, equal to 1 kg m s − , and the unit of energy the joule, equal to 1 N m= 1 kg m s − . The unit of angular momentum, or action, is the kg m s − , or J s. But the quantity ofmatter of the international prototype of the kilogram cannot remain exactly constant, and varies veryslightly over the years. In fact the differences in mass between the IPK and supposedly identical copiesaverage to a few tens of µ g per century [9]. Could we avoid having to resort to such a physical object todefine the unit of mass, and relate it instead to a reproducible universal phenomenon, as for the secondand the metre ? This is where quantum physics comes in, as angular momenta and actions are quantizedin units of ~ = h/ π , whose value in SI units is ≃ . ... × − J s .
VI. QUANTUM PHYSICS AND THE CORRESPONDENCE PRINCIPLE
Quantum physics allows to relate energies with frequencies, and momenta with wavelengths. It involvesthe Planck constant h , which relates, through E = hν = ~ ω , the energy E carried by a photon withthe frequency ν = ω/ π of the corresponding electromagnetic wave, and similarly for the other particles.A particle of momentum ~p is associated with a wave of wave vector ~k given by ~p = ~ ~k , and wavelength λ given by the De Broglie relation p = ~ k = h/λ , where ~ = h/ π is the reduced Planck constant.Quantum mechanics thus relates, through the constants h or ~ , energies with inverses of times, andmomenta with inverses of lengths, E ∝ t − , p ∝ l − . This is an expression of the correspondenceprinciple , which associates with each particle (including the photon as a quantum of light) of energy E and momentum ~p a wave ψ ( ~x, t ) ∝ e i ( ~k.~x − ωt ) = e − i ( Et − ~p.~x ) / ~ , with E = hν = ~ ω , ~p = ~ ~k . (28)It may be formulated in a Lorentz-invariant way in terms of the spacetime coordinate x µ = ( ct, ~x ), as ψ ( x µ ) ∝ e − i k µ x µ = e − i p µ x µ / ~ , with p µ = ( E/c, ~p ) = ~ k µ = ~ ( ω/c, ~k ) . (29)Thus a measurement of energy, or momentum, can be replaced by an associated measurement for thecorresponding angular frequency, or wave number, once the numerical value of ~ (in J s) is experimentallydetermined. For a free particle of mass m , p µ x µ = mc τ where τ is the proper time experienced by theparticle, so that ψ ( x µ ) in (29) may also be expressed as a function of the proper time, proportionally to e − i mc τ/ ~ .Even better, a measurement of energy (or momentum) becomes identical to a measurement of angularfrequency (expressed in s − ) or wave number (expressed in m − ), if the value of ~ gets fixed, in agreementwith its present best experimental determination, within uncertainties. Units of energies or momenta getthen derived from the corresponding units of angular frequencies or wave numbers. The present definitionof the kilogram, taken since 1889 as the mass of the international prototype stored at the BIPM, is nolonger necessary, and may be abandoned in favor of a new definition based on quantum physics. VII. DEFINING THE JOULE AND KG FROM THE INVERSE OF THE SECOND,THROUGH QUANTUM MECHANICS ~ is the fundamental quantum of action, or angular momentum, in quantum mechanics. It has thedimensions of (energy × time), or (momentum × space), or angular momentum ( × angle, dimension-less). It is measured in J s, or equivalently kg m s − , as also seen from the expressions of the operatorshamiltonian, momentum, and orbital angular momentum, H = i ~ ∂∂t , ~P = − i ~ ∂∂~x , L z = − i ~ ∂∂ϕ , (30)in agreement with the correspondence principle. In particular the plane waves ψ ∝ e i ( ~k.~x − ωt ) in (28) are0eigenfunctions of H and ~P with the eigenvalues E = ~ ω and ~p = ~ ~k . They are also eigenfunctions of theoperator i ~ c ddτ where τ is the proper time experienced by the particle, with its mass m as eigenvalue.We make no difference between the so-called “inertial” and “gravitational” masses of a particle ormacroscopic object, m i and m g . They may be taken as identical, as investigated long ago by E¨otv¨osand his collaborators. [17]. The resulting Equivalence Principle, formulated by Einstein, is at the basis ofGeneral Relativity, which provides the theory of gravitation, at the classical level. The MICROSCOPE experiment provides at present the most stringent test on the validity of this principle, allowing for theidentification of inertial and gravitational masses at a level of precision of 2 × − , for the pair ofmaterials tested [18].The reduced Planck constant, ~ = h/ π , is such that the intrinsic angular momentum of the electron,or spin, is ~ /
2. The electron is a spin-1/2 particle, meaning that its spin is S = 1/2, when ~ is taken equalto 1 . Fixing ~ = 1 also defines the unit of energy from (or even identical to) the unit of angular frequency,namely the s − (in principle × ~ , which is 1). The expression of ~ (close to 1 .
054 571 818 × − J sif h = h ◦ J s is fixed at 6.626 070 15 × − J s ), implies that the natural unit of energy is unit of energy u e = ~ s − = ~ ◦ J = 6 .
626 070 15 × − π J = 1 .
054 571 817 646 ... × − J (31)(or kg m s − ). Fixing c = 1 also determines, as in (23), the unit of length u l = c s = c ◦ m as thedistance travelled by light in vacuum during the unit of time, here the second. This ultimately allows foridentifying the unit of length with the unit of time, and thus to measure distances directly in seconds.The unit of mass is then the same as for energy (as c = 1), namely u m = u e = 1 s − . This single unitmay be reexpressed in conventional units using the joule as unit of energy or the kg as unit of mass, sothat unit of mass u m = ~ /c s − = ~ ◦ /c ◦ kg = 1 .
173 369 392 016 ... × − kg . (32)The unit of angular momentum, now dimensionless, is 1. This may be verified from u m = ( ~ /c ) s − , u l = c s , u t = 1 s, combining into unit of angular momentum u j = u m u l /u t = ~ . It may be reexpressed, inconventional units, as ( h ◦ / π ) J s = 1 .
054 571 817 646 ... × − J s .For a theoretician, measuring space directly in seconds and masses and energies in s − would be thenatural thing to do. This is not very practical however, the second as a unit of distance being too large,while the s − as a unit of energy (close to 10 − J) or unit of mass (close to 10 − kg) is too small.The unit of angular momentum when ~ = 1, now the dimensionless number 1, is suitable for individualphotons, electrons or nucleons, but too small for macroscopic objects.It is thus convenient to resize consistently the above units (23,31,32) of length, energy, and mass toturn them into more convenient ones, still keeping the familiar names of metre, joule, and kilogram inorder not to disrupt long habits and use easily former measurements. The quantities c and h or ~ , oftenpreviously referred to as “fundamental constants of nature”, get fixed, according to the resolution [10]“ the speed of light in vacuum c is 299 792 458 m/s ”,“ the Planck constant h is 6.626 070 15 × − J s ”. (33)and used as normalization constants in the redefinition of the system of units.Just as we can use the above choice of c to redefine the metre from the natural unit of length u l as in(23), we can use the choice of a fixed value of h (or ~ ), namely h = h ◦ J s, to redefine the joule and thekilogram from the natural units and energy and mass in (31,32). This leads to /c ◦ unit of length u l , / ~ ◦ unit of energy u e , c ◦ / ~ ◦ unit of mass u m , (34)which define the metre, the joule, and the kilogram.1 VIII. FIXING h = 6 .
626 070 15 × − J s , AND ~ = 1 Beyond that, we can identify u l = u t = 1 s, and u m = u e = 1 s − . This leads to identify the metre, thejoule, the kilogram, and the newton, etc., as fixed numbers of s, s − or s − , etc.. Indeed, beyond the officialdefinitions c = c ◦ m s − with c ◦ fixed at 299 792 458, h = h ◦ J s with h ◦ fixed at 6.626 070 15 × − , wenow identify ( c = c ◦ m s − ≡ , ~ = ~ ◦ J s ≡ , = ⇒ ~ /c = ~ ◦ /c ◦ kg s ≡ , ~ /c = ~ ◦ /c ◦ kg m = ~ ◦ /c ◦ N s ≡ , ~ c = ~ ◦ c ◦ J m ≡ . (35)This leads to the identifications1 metre = 1 c ◦ s , ~ ◦ s − , c ◦ ~ ◦ s − , c ◦ ~ ◦ s − , (36)or, explicitly, /
299 792 458) s , π .
626 070 15 × − s − = .
948 252 156 246 ... × s − , π × (299 792 458) .
626 070 15 × − s − = .
852 246 536 175 ... × s − , − = 1 J m − = 2 π ×
299 792 4586 .
626 070 15 × − s − = 2 .
842 788 447 250 ... × s − . (37)We can then say that the electron is a spin-1/2 particle, with ~ = ~ ◦ J s = 1, the speed of light being c =299 792 458 m/s, and, at the same time, c = 1 , with µ ◦ = µ ◦ N/A = 1. Should we fix h , or rather ~ ? The above equations (34-37) also indicate that, at least from a conceptual point of view, it could havebeen preferable to fix numerically the value of ~ (possibly to 1.054 571 82 × − J s), which determinesdirectly the angular momentum of the photon ( ~ ), or of the electron ( ~ / h . Thiswould also have led to simpler definitions for the joule, the kilogram, and the newton in (34-37).Anyhow, once such a choice has been made, the definitions of the joule, and of the associated kilogram,can also be formulated through one of the sentences the angular momentum of an electron is
12 6 .
626 070 15 × − π J s (or kg m s − ) , the angular momentum of a circularly polarized photon is .
626 070 15 × − π J s (or kg m s − ) , (38)which give a direct physical meaning to the fixing of Planck’s constant h in (1,33).Still, defining in this way the metre, the joule, the kilogram, and the newton from the second, thes − , or the s − also requires a mise en pratique allowing one to apply these definitions to macroscopicobjects. The Kibble balance, in particular, appeals to electrical measurements relying on electromagneticinteractions in the quantum regime, implying the Josephson and von Klitzing constants K J = 2 e/h and R K = h/e [19]. One can also count a larger number of atoms in a crystal, and determine the mass of asilicon sphere as m = (8 V /a ) m (Si), where m (Si) is the mean mass of a silicon atom in the crystal, and a the volume of the unit cell, with eight atoms on average [20].2 The redefinition of the kilogram from quantum mechanics does not rely on interactions
The definition of the kilogram from quantum mechanics through fixing the value of h or ~ , however,does not in principle appeal to interactions, and in particular does not require the explicit consideration ofelectromagnetic interactions. This is true even at the macroscopic level, as we shall see with the Casimirforce, which depends only on ~ and c , and whose expression is fixed if once ~ c is fixed.Indeed the momentum of a particle, and thus its mass, or the energy of a photon, may be determinedfrom the dimension associated with the diffraction or interference pattern associated with it. This involvesonly free particles. Indeed the Dirac equation for a free spin-1/2 field or particle, the Klein-Gordonequation for a free spin-0 field or particle, and the Maxwell equations for a free massless spin-1 field,expressed as Dirac : ( i ~ γ µ ∂ µ − mc ) ψ = 0 , Klein-Gordon : ( ~ ∂ µ ∂ µ + m c ) ϕ = 0 , Maxwell : ∂ µ F µν = 0 , (39)involve the constants c and ~ associated with the definition of the system of units, but not the interactions.With c = ~ = 1 (a choice compatible with c = 299 792 458 m/s and h = 6.626 070 15 × − J s as wesaw), these equations get simply written as( i ∂/ − m ) ψ = 0 , ( ✷ + m ) ϕ = 0 , ∂ µ F µν = 0 . (40) These fundamental equations no longer involve the “fundamental constants” c and ~ ! But their numericalvalues in SI units, c ◦ and ~ ◦ , now serve in the definition of the fundamental units of length and mass.The consideration of the Casimir force between two conducting plates allows in principle for passingfrom the microscopic to the macroscopic level for the measurement of a force and thus also a mass,defined as earlier by fixing h or ~ . Ideally the Casimir force depends only on geometry, and associates toa distance l between two conducting plates a force per unit surface, i.e. a pressure. Its expression involves c and ~ . Once c is fixed it may be used, in principle, to determine the value of ~ in ordinary units of J s,even if only with a very modest precision. Or conversely, if ~ and thus ~ c is fixed, it allows to realize thejoule from the metre. Considered as a force per unit surface it reads P = FS = − π ~ cl ≃ − .
013 dyn/cm l ( µ m) ≃ . l ( µ m) . (41)It indicates how a very small force might be realized experimentally at the macroscopic level, even if notprecisely, once ~ is fixed and the unit of length has been defined by fixing c . This allows us, in principle,to pass at the macroscopic level from a geometric unit of distance to a mechanical unit of force and,subsequently, energy and mass, without having to consider explicitly the value of the elementary charge e (but taking into account boundary conditions for the electromagnetic field between the plates).In practice the precision of quantum electromagnetic effects is essential to get a precise determinationof h through K J = 2 e/h and R K = h/e , and taking full advantage of fixing h [19]. This motivates amore precise definition of electrical units than the usual one through the traditional definition of theampere [10, 14, 21]. IX. ELECTRICAL UNITS AS TIED TO THE MECHANICAL ONESA. The ampere and the other electrical units, as obtained from µ ◦ The electrical units are at the moment rigidly tied to the mechanical ones through a fixed µ ◦ as seenin Section II. Since 1946 and up to now in 2018, the ampere A has been defined as follows [12, 13] : “ The ampere is that constant current which, if maintained in two straight parallel conductorsof infinite length, of negligible circular cross-section, and placed apart in vacuum,would produce between these conductors a force equal to × − newton per metre of length.” (42)3This originates from Amp`ere’s force law stating that the force F per length L between two such parallelconductors at distance r , through which a current of intensity I passes, is F/L = ( µ ◦ / π ) I /r . Thisdefinition involves the fixed number 2 × − associated with the factor µ ◦ , referred to as the magneticpermeability of free space, conventionally taken as 4 π × − N/A in the present SI (but soon to become4 π × − η N/A ).Amp`ere’s force law, however, is a fundamental law of physics, and it seems more logical to writeit independently of the somewhat arbitrary parameter µ ◦ , which determines the sizes chosen for theelectrical units (very much as done for eqs. (40) when writing the free equations of motions independentlyof c and ~ ). We shall thus discard the factor µ ◦ from its expression (or write it with an implicit parameter µ ◦ = 1), to include it as a numerical dimensionless coefficient (4 π × − in the present SI) within theman-made definition of the ampere. This allows for a new equivalent formulation, in which 1 A = µ ◦ N ,the ampere being obtained as proportional to a square root of the newton as in (2),
Amp`ere’s force law FL = I π r = ⇒ p µ ◦ N . (43)This provides the force per unit length between two conductors carrying currents of 1 A at a distanceof 1 metre in vacuum, obtained from (43) as F = 1 A / π = ( µ ◦ / π ) N, i.e. at the moment, before theredefinition of the SI, as 2 × − N as in (42).Writing as above fundamental laws with µ ◦ = 1, the ampere, the coulomb, and the volt get given bythe new expressions p µ ◦ N , p µ ◦ kg m , / C = 1 W / A = 1 √ µ ◦ W/ √ N = s J m s − µ ◦ . (44)The coulomb, in particular, is proportional to a geometric mean of the kilogram and the metre .The definitions (44), and (45) below, of the various electrical units follow, in agreement with the( µ ◦ -independent) expressions for electrostatic energy ( qV ) or electrical power ( U I ) for the volt ; Laplace’smagnetic force on a wire ( ~dF = I ~dl × ~B ) for the tesla ; energy stored in a coil or capacitor ( LI / CV /
2) for the henry and farad ; and power dissipated into a resistance ( RI ) for the ohm. A magneticfield also appears as proportional to the square root of the corresponding energy density (expressed as B /
2, with µ ◦ = 1). An inductance is proportional to a length and a resistance to a velocity, etc., with / (A m) = 1 √ µ ◦ q N/m = 1 √ µ ◦ q J/m , / A = 1 µ ◦ J/N = 1 µ ◦ m , / V = 1 J / V = µ ◦ s / m , / A = 1 W / A = 1 µ ◦ W/N = 1 µ ◦ m/s . (45)The resulting expressions of electrical units are given in Table I.We note the usual relations 1 H = 1 Ω s , / Ω , (46)in agreement with the characteristic times τ = L/R of an inductance and τ = RC of a capacitor inassociation with a resistance (with 1 H × , as from the relation LCω = 1 for an LC oscillator).These definitions and relations are by construction compatible with the rescaling of µ ◦ by a factor η ,acting on the various units as in (18).4 B. Fixing c = c ◦ m/s, ~ = ~ ◦ J s, then also c = ~ = 1 We can then take advantage of fixing c = c ◦ m / s = 1 , ~ = ~ ◦ J s = 1, to express the ampere, and allelectrical units, in terms of the second, s − or s − , or even as fixed numerical constants, as displayed inTable I of Section II. An impedance, Z = R + j ( Lω − /Cω ), gets dimensionless, and may be expressed inohms, a dimensionless unit, equal to 1 /µ ◦ c ◦ , with the “impedance of vacuum” being now Z ◦ = µ ◦ c ◦ Ω = 1.To understand things better we decompose the fixing of c and ~ in two steps, first by fixing their valuesin m/s and J s as in the new SI [10], then by choosing also c = ~ = 1 as proposed here. With c = c ◦ m/s, ~ = ~ ◦ J s, we express the ampere and coulomb as √ µ ◦ N = s µ ◦ J sm / s s − = r µ ◦ c ◦ ~ ◦ r ~ c s − , √ µ ◦ kg m = s µ ◦ J sm / s = r µ ◦ c ◦ ~ ◦ r ~ c . (47)Electric charges, including the coulomb, still have a dimension, and may be evaluated at this stage in √ kg m, with µ ◦ taken as dimensionless.With the additional choice c = ~ = 1, the scale for all electrical units gets fixed (in terms of µ ◦ ) andformulas (47) simplify into √ µ ◦ N = r µ ◦ c ◦ ~ ◦ s − = 1 . ... × s − , √ µ ◦ kg m = r µ ◦ c ◦ ~ ◦ = r ǫ ◦ c ◦ ~ ◦ = 1 . ... × , (48)the coulomb becoming a dimensionless unit. Electrical charges are now dimensionless, as for angles, theirnatural unit being 1 in a system with ~ = c = µ ◦ = ǫ ◦ = 1. The coulomb is an artificial unit, resultingfrom the definition (42) of the ampere (from µ ◦ ), and of the metre and joule through the choices of c ◦ and ~ ◦ . The elementary charge e , now dimensionless, is expressed (as seen from (15)) as e = e ◦ C = p e ◦ /ǫ ◦ c ◦ ~ ◦ = √ πα = √ πα ◦ η = .
302 822 120 789 ... × η , (49)thanks to the choice ~ = c = µ ◦ = ǫ ◦ = 1. The numerical value of e is thus rescaled by the factor η = p α/α ◦ , very close to 1.At the same time, with 1 Ω = 1 W / A = (1 /µ ◦ ) m / s = 1 /µ ◦ c ◦ = ǫ ◦ c ◦ , the impedance of vacuum is Z ◦ = µ ◦ c ◦ Ω = 376 .
730 313 461 ... × η Ω = 1 , (50)as it should with c = µ ◦ = ǫ ◦ = 1 . (The above numerical value in ohms has been evaluated with µ ◦ = 4 π × − , to be rescaled by η , with Z ◦ remaining equal to 1.) The impedance being now adimensionless quantity its natural unit, 1, is simply what is also referred to as the “impedance of vacuum”.From (45) and using c = c ◦ m/s = 1, we find, with 1 F = µ ◦ c ◦ m = (1 /ǫ ◦ ) m, the now symmetricexpressions for the farad and the henry, given by µ ◦ H = ǫ ◦ F = 1 m , (51)or equivalently µ ◦ c ◦ H = ǫ ◦ c ◦ F = 1 s . (52)5With 1 Ω = 1 /µ ◦ c ◦ = ǫ ◦ c ◦ we recover eqs. (46),1 H / Ω = 1 F Ω = 1 s . (53)For the tesla and V/m, we get from (44,45) the analogous expressions √ µ ◦ q J/m , / m = 1 √ µ ◦ q J/m m / s = √ ǫ ◦ q J/m = 1 c ◦ T , (54)thanks in the last case to the identification 1 m = (1 /c ◦ ) s. This will be interpreted in subsection X B interms of the electric and magnetic field energy densities, now simply given by E / B /
2, corres-ponding to their standard SI expressions as E = ǫ ◦ E / B / µ ◦ , respectively. C. A direct characterization of the coulomb
Coulomb’s law is now written with ǫ ◦ = 1 as F = q / πr . With 1 A = √ µ ◦ N so that 1 C = p µ ◦ c ◦ N m= p N /ǫ ◦ m, one has 1 C = 1 N m /ǫ ◦ . The force between two charges of 1 C at 1 metre apart in vacuumis recovered as (1 / πǫ ◦ ) N. We can duplicate (43) for the ampere, writing the parallel expression for thecoulomb, Coulomb’s law F = q πr = ⇒ r N ǫ ◦ m . (55)This provides a physical understanding for the coulomb. The electrostatic force between two charges q = 1 C, at r = 1 m apart in vacuum, is obtained as (1 / πǫ ◦ ) N = ( µ ◦ c ◦ / π ) N = 10 − c ◦ η N =8 .
987 551 787 ... × N. Their electrostatic energy V = µ ◦ c q / πr is E = µ ◦ π c ◦ J = 10 − η (299 792 458) J = 8 .
987 551 787 ... × η J . (56)This energy is no longer exactly known, owing to the new definition of the coulomb. The equivalent mass,according to relativity, is Ec = 1 c q πǫ ◦ r = µ ◦ π q r = 10 − η q r . (57)The coulomb may be characterized as follows :“ The coulomb is such that the electrostatic energyof two charges of , apart in vacuum, is equal to (1 / πǫ ◦ ) J .” (58)or : “ The coulomb is such that the mass equivalent to the electrostatic energyof two charges of , apart in vacuum, is equal to µ ◦ / π ( i.e. 10 − η ) kg .” (59)The factor η , at the moment equal to 1 within the present SI, takes into account that the numerical valueof the elementary charge e is getting redefined as in (7), with µ ◦ / π very slightly shifted away from 10 − ,proportionally to η . X. PROPERTIES OF THE NEW SYSTEMA. Invariance of the fine structure constant α under a rescaling of the fundamental units If we reintroduce explicitly the constant µ ◦ = µ ◦ N / A (momentarily considered as non-necessarilyequal to 1), the ampere and coulomb become expressed as1 A = q ( µ ◦ /µ ◦ ) N , q ( µ ◦ /µ ◦ ) kg m . (60)6With ~ = ~ ◦ J s, c = c ◦ m/s, so that ~ /c = ( ~ ◦ /c ◦ ) kg m, we get, for the elementary charge, e = e ◦ C = µ ◦ e ◦ µ ◦ ~ /c ~ ◦ /c ◦ , so that µ ◦ c e ~ = µ ◦ c ◦ e ◦ ~ ◦ , (61)or α = e πǫ ◦ ~ c = e ◦ πǫ ◦ ~ ◦ c ◦ . (62)This provides an identity between the two expressions of the fine structure constant, associated, for anyset of values of ~ , c, µ ◦ (and corresponding ǫ ◦ ), with the rescaling of the fundamental units through thechoice c = c ◦ m / s , ~ = ~ ◦ J s , µ ◦ = µ ◦ N / A .If as a further step we choose ~ = c = µ ◦ = ǫ ◦ = 1, we recover α = e π = e ◦ πǫ ◦ ~ ◦ c ◦ (63)leading to e = √ πα = . ... as in (49). B. Invariance of the impedance of vacuum Z ◦ under a rescaling of the fundamental units Let us return to the SI for a short moment. The energy density associated with an electromagneticfield, E = ǫ ◦ E B µ ◦ = E + c B µ ◦ c = r ǫ ◦ µ ◦ (cid:18) E c + cB (cid:19) . (64)with ǫ ◦ µ ◦ c = 1, originates from the Lagrangian density L = − (1 / µ ◦ ) F µν F µν . The quantity Z ◦ = r µ ◦ ǫ ◦ = µ ◦ c (65)equal to the ratio E/ ( B/µ ◦ ) for a plane electromagnetic wave in vacuum, is referred to as the impedanceof vacuum. It appears as a normalisation coefficient in Mawxell’s equations ~ rot ~B = µ ◦ c ( ~j/c ) + ... ,div ~E/c = ρ/ǫ ◦ c , which fix the relative scale of ( ~E/c, ~B ) compared to that of the current density ( ρ, ~j/c ),source of the electromagnetic field. This may also be seen from Poisson’s equation for a static field,expressed as ∆( V /c, ~A ) = − µ ◦ c ( ρ, ~j/c ), with div ~A = 0.With µ ◦ expressed in N/A or H/m and c in m/s, Z ◦ is obtained in the SI in Ω. We have at present Z ◦ = µ ◦ c = (4 π × − H / m) ( c ◦ m / s) = 4 π × − c ◦ Ω = 376 . ... Ω . (66)But if the elementary charge gets fixed as in (7), µ ◦ gets changed into 4 π × − η . The measure in ohmsof the impedance of vacuum is multiplied by η , to become in the new SI Z ◦ = 4 π × − η c ◦ Ω = 376 . ... η Ω . (67)But this does not mean that the impedance of the vacuum has been multiplied by η ! At the same time,with 1 Ω = 1 W/A , the size of the ohm is divided by η as in (18), according toΩ = Ω old /η , (68)so that Z ◦ itself, equal to µ ◦ c ◦ Ω, remains unchanged. As for the elementary charge e , expressed eitheras e old C old or as e ◦ C, Z ◦ , which is an intrinsic quantity, may be expressed equally well as Z ◦ = 4 π × − c ◦ | {z } old = 4 π × − η c ◦ Ω . (69)This ensures that the impedance of vacuum Z ◦ (but not its measure in ohms !), is indeed insensitiveto the change in electrical units following from the fixing of e = e ◦ C within the new SI, and remains7insensitive to the value of the elementary charge e – i.e. to the value of α to be measured experimentallyin the future. This is fortunate, as Z ◦ is an intrinsic quantity that should not depend on the chosensystem of units, nor on future measurements of α . But it also illustrates how fixing the numerical valueof the electrical charge in the new SI as in (7), while practically convenient, makes life more complicatedas far as explaining what the impedance of the vacuum really is, and for which reason its measure shouldnow depend on α .In the new proposed system their is no such concern, as 1 Ω = 1 /µ ◦ c ◦ and Z ◦ = µ ◦ c ◦ Ω = 1, indepen-dently of any rescaling of µ ◦ in agreement with (18). This is a consequence of ~ = c = µ ◦ = ǫ ◦ = 1 sothat the impedance of vacuum is the unit of impedance, namely simply 1. The Lagrangian and energydensities for the free electromagnetic field get simply expressed as L = − F µν F µν , E = E + B , (70)as usual in relativistic quantum field theory, and its equation of motion reads ∂ ν F µν = j ν . C. The volt/m, the tesla and the electromagnetic energy
Expressions (54) of the tesla and volt/metre read1 T = 1 √ µ ◦ q J/m , / m = √ ǫ ◦ q J/m = 1 c ◦ T . (71)We reobtain correctly the energy densities for unit magnetic and electric fields of 1 T and 1 V/m, obtainedfrom (70) as E = B µ ◦ J / m , and E = E / m) ǫ ◦ / m . (72)This also provides a simple characterization of the tesla and volt/metre, as follows : “ The tesla is the magnetic field corresponding to an energy density in vacuum / µ ◦ (or 10 / πη ) J/m . ”“ The volt/metre is the electric field corresponding to an energy density in vacuum ǫ ◦ / / πc ◦ η ) J/m . ” (73)The energy density for a magnetic field of of 1 T is larger than for an electric field of 1 V/m by a largefactor 1 /µ ◦ ǫ ◦ = c ◦ ≃ × , as easily understood since 1 T = c ◦ V/m.Let us also consider the Poynting vector, now simply expressed as ~P = ~E × ~B . For orthogonal unitelectric and magnetic fields of 1 V/m and 1 T, the energy flux density per unit time would be1 V / m × p c ◦ /µ ◦ ~ ◦ s − × p c ◦ /µ ◦ ~ ◦ s − = 1 µ ◦ W / m = 14 π × − η W / m , (74)with c ◦ s − = 1 m − , and 1 / ~ ◦ = 1 J s. This is evaluated in a more conventional way in the SI with ~P = ~E × ~B/µ ◦ , as 1 V / m × µ ◦ = 1 (W / A m) (N / A m)4 π × − η N / A = 14 π × − η W / m . (75)Eq. (74) illustrates how the factor 4 π × − η , instead of being present in the expression of the Poyntingvector as in the standard formulation, gets now included within the new expressions of the electrical unitsas in (71).Electrical units thus remain strongly tied to mechanical ones and completely fixed by them, up to thescale factor µ ◦ = 4 π × − η , where η is very close to 1. These formulas reflect the tight connectionbetween electrical and mechanical units, originating from the traditional definition of the ampere. Thisconnection gets somewhat weakened as µ ◦ is no longer fixed if the value of the elementary charge getsfixed as in (7). Electrical units are then no longer rigidly tied to mechanical ones, but in a more flexibleway, proportionally to √ µ ◦ i.e. to η = p α/α ◦ for the ampere and the coulomb, as described by eqs. (18).8 D. Inductances and capacitances
The henry may be defined from the magnetic energy stored in a coil, E = LI /
2, so that1 H = 1 J / A = 1 µ ◦ J / N = 1 µ ◦ m = 1 µ ◦ c ◦ s = 1(376 . ... η ) s . (76)The inductance L for a long coil of length l , inner area S and N turns is now simply given, with µ ◦ = 1,by L = N S/l . For a coil of unit length l = 1 m the inductance per unit inner core area S (supposedsmall) and unit N is L = 1 m = µ ◦ H , (77)as found with the usual SI formula L = µ ◦ N S/l . Similarly, the capacitance of a plane capacitor withempty space between the plates is given, with ǫ ◦ = 1, by C = S/l . For a unit distance between the plates l = 1 m, the capacitance per unit surface S is C = 1 m = ǫ ◦ F , (78)as seen from (51), and as found with the usual SI formula C = ǫ ◦ S/l . The impedance p L/C evaluatedwith the above values L = 1 m = µ ◦ H and C = 1 m = ǫ ◦ F is the impedance of free space, which is 1 : p L/C = p µ ◦ H /ǫ ◦ F = µ ◦ c ◦ p H / F = µ ◦ c ◦ Ω = Z ◦ = 1 . (79) E. The weber and the quantum of flux
The weber is the magnetic flux induced by a current of 1 A circulating in a coil of inductance L = 1 H,1 Wb = 1 H × / p µ ◦ c ◦ ~ ◦ = 1 V s = 5 .
017 029 284 119 ... × η − . (80)It is a dimensionless unit, as for the coulomb and the ohm, equal to 1 V s (as 1 H = 1 Ω s). These threedimensionless units, p µ ◦ c ◦ / ~ ◦ = 1 . ... × η , /µ ◦ c ◦ = 1 / (376 . ... η ) , / √ µ ◦ c ◦ ~ ◦ = 5 . ... × η − , (81)satisfy the relation, independent of c ◦ , µ ◦ and ~ ◦ ,1 C × , (82)reflecting that 1 A × × ◦ = h e = π ~ ◦ e ◦ J s / C | {z } Wb = π s ~ ◦ µ ◦ c ◦ e ◦ = r π α = πe ≃ .
374 382 97 , (83)or Φ ◦ = h/ e = 2 .
067 833 848 461 ... × − Wb = πe ≃ .
374 382 97 . (84)We simply recover π/e with e = √ πα , at no surprise since ~ = c = µ ◦ = ǫ ◦ = 1. The inverse of the fluxquantum is the Josephson constant, K J = 1Φ ◦ = 2 eh = 2 e ◦ h ◦ Wb − = 483 597 .
848 416 ...
GHz / V = eπ ≃ . . (85)It is related to the expression of 1 eV = π (2 e ◦ /h ◦ ) s − = 1 . ... × s − , as we shall see in (91-93).The expression of the physical laws, now written without reference to ~ , c, µ ◦ and ǫ ◦ , and the newexpressions for the corresponding units, are given in Table II.9 Table
II: The physical laws written in a universal way, with no reference to ~ , c, µ ◦ and ǫ ◦ . This includesMaxwell’s equations. ~ ◦ , c ◦ , µ ◦ and ǫ ◦ define our usual units by scaling appropriately the natural ones obtainedfrom the second. The usual formulas including µ ◦ and ǫ ◦ are shown in the last column. Due to the fixing of e = e ◦ C, µ ◦ can no longer be fixed at 4 π × − but should be multiplied by η = p α/α ◦ , which is very close to1 (cf. eqs. (14-16)).Physical lawor expression New expression Usual expressionAmp`ere’s force law FL = I πr with 1 A = p µ ◦ N ←→ FL = µ ◦ I πr Coulomb’s law F = q πr ” 1 C = r N ǫ ◦ m ←→ F = q πǫ ◦ r electric energy density E = E / m = p ǫ ◦ J / m ←→ E = ǫ ◦ E E = B s J / m µ ◦ ←→ E = B µ ◦ Poynting vector ~ P = ~E × ~B ” 1 (V / m) T = W / m µ ◦ ←→ ~ P = ~E × ~Bµ ◦ Lagrangian density L = − F µν F µν ←→ L = ǫ ◦ E − B µ ◦ cap. of plane capacitor C = Sl ” 1 m = ǫ ◦ F ←→ C = ǫ ◦ Sl inductance of coil L = N Sl ” 1 m = µ ◦ H ←→ L = µ ◦ N Sl impedance of vacuum Z ◦ = 1 ←→ Z ◦ = r µ ◦ ǫ ◦ Ω ≃
377 Ωfine structure constant α = e π = e ◦ π C ” 1 C = r ǫ ◦ c ◦ ~ ◦ ←→ α = e ◦ πǫ ◦ ~ ◦ c ◦ ≃ F. Relating energies with distances and times
The spectrum of the hydrogen atom, for example, may be obtained with ~ = c = ǫ ◦ = 1 and e = √ πα ,which leads directly to the Rydberg energy1 Ry = 12 m e ( e / π ) = 12 m e α ≃ .
605 6930 eV . (86)We can use the fine structure constant α = e / π ≃ / .
035 9991 ... , and express the mass of theelectron as m e ≃ .510 998 946 MeV, with no need to refer to the velocity of light.A useful formula to relate energies and distances is obtained by evaluating the product 1 eV × ~ ◦ J s) ( c ◦ m / s) = ~ ◦ c ◦ e ◦ eV m = 10 ~ ◦ c ◦ e ◦ MeV fm , (87)using10 (6 .
626 070 15 × − ) × (2 .
997 924 58 × )2 π × .
602 176 634 × − = 197 .
326 980 459 302 ... (now exactly known) .(88)0This may be remembered, not surprisingly, as 197. 327 MeV Fermi ≃ − = 197 .
326 980 459 ... fm . (89)To give some illustrative examples, the Lyman α wavelength, Bohr radius, reduced Compton wavelengthof the electron, range of weak interactions and Planck length may be expressed directly using (89), withno reference to ~ and c factors, as Lyman α wavelength ( ≈
43 (1 + m e m p ) 2 π .
606 eV ) ≃ .
67 ˚A ,r B = 1 m e α ≃ .
035 999 .
510 998 946 MeV ≃ .
529 177 21 ˚A ,λ e / = 1 m e ≃ .
510 998 946 MeV ≃ .
861 5927 × − m ,λ W / = 1 m W ≃ .
38 GeV ≃ . × − m ,l P = 1 m P = p G N ≃ . × GeV ≃ . × − m . (90)The relation between energy and time comes from 1 J = (1 / ~ ◦ ) s − = .
948 252 156 ... × s − , so that1 eV = e ◦ J = e ◦ ~ ◦ s − = 1 .
519 267 447 878 626 ... × s − = 1 / (6 .
582 119 569 509 065 ... × − s) . (91)also equivalent to1 MeV − = 197 .
326 980 459 302 ... fm = 6 .
582 119 569 509 ... × − s . (92)1 eV s/ π = 2 e ◦ /h ◦ = .
483 597 848 416 983 ... × involves the same factor as for K J = (2 e ◦ /h ◦ ) Hz/V.We have the equivalence between the new expression of K J as equal to e/π , and the well-known expressionof ~ , precisely evaluated as ( ~ ◦ /e ◦ ) eV s = 6 .
582 119 569 509 ... × − MeV s, and now equal to ~ = 1 : K J = .
483 597 ... × Hz / V = e/π ⇐⇒ π × .
483 597 848 416 ... × s − = 1 .
519 267 447 878 ... × s − . (93) G. The kelvin and the Boltzmann constant
In the new SI, the unit of thermodynamic temperature, the kelvin K, will be derived from the unit ofenergy by fixing the numerical value k ◦ of the Boltzmann constant k [10], so that k = 1 .
380 649 × − J / K . (94)We can then go one step further. Very much as space and time are related and may both be measuredin seconds, or as energy may be measured in s − when time is measured in seconds, thermodynamictemperature and energy are related and may both be measured with the same unit, the s − , by choosinga unit value of the Boltzmann constant k . The Maxwell-Bolzmann distribution, for example, will thenbe simply expressed with k = 1, proportionally to e − E/T .Choosing k = 1 is compatible with fixing its value in SI units according to (94), leading to k = 1 .
380 649 × − J / K = 1 . (95)This allows to identify the kelvin K with a certain number k ◦ of joules according to1 K = 1 .
380 649 × − J . (96)1We can also express energies in eV, rather than in joules. We then have the exact relation1 eV = 1 .
602 176 634 × − J = 1 .
602 176 634 × − .
380 649 × − K = 11 604 .
518 121 ... K , (97)remembered as 1 eV ≃
11 605 K.
H. A more symmetric treatment between electricity and magnetism
In this discussion, and within the SI, µ ◦ has been given a favored treatment as compared to ǫ ◦ , owing tothe original definition of the ampere. But there is no special reason to do so, as they contribute in similarways to the expressions of the speed of light c = 1 / p ǫ ◦ µ ◦ and impedance of vacuum Z ◦ = q µ ◦ / ǫ ◦ .To get a system with c = µ ◦ = ǫ ◦ = 1, as proposed here, we may use, equivalently, two of the four setsof conditions c = c ◦ m / s = 1 ,Z ◦ = z ◦ Ω = 1 ,µ ◦ = µ ◦ H / m = µ ◦ c ◦ H / s = 1 ,ǫ ◦ = ǫ ◦ F / m = ǫ ◦ c ◦ F / s = 1 , (98)where the last three are associated with eqs. (50-52). For example we already have the equivalences ( c = c ◦ m / s ,Z ◦ = z ◦ Ω , ⇐⇒ µ ◦ = Z ◦ c = z ◦ c ◦ Ω sm = µ ◦ H / m = µ ◦ c ◦ H / s ,ǫ ◦ = 1 c Z ◦ = 1 c ◦ z ◦ sΩ m = ǫ ◦ F / m = ǫ ◦ c ◦ F / s , (99)using relations (46) between units, 1 Ω s = 1 H, 1 s/Ω = 1 F, together with ( c ◦ = 1 / √ µ ◦ ǫ ◦ ,z ◦ = p µ ◦ /ǫ ◦ , ⇐⇒ ( µ ◦ = z ◦ /c ◦ ,ǫ ◦ = 1 /c ◦ z ◦ . (100)This leads to the equivalence between the two sets of conditions in (98), where electricity and magnetismplay similar roles, ( c = c ◦ m / s = 1 ,Z ◦ = z ◦ Ω = 1 , ⇐⇒ ( µ ◦ = µ ◦ c ◦ H / s = 1 ,ǫ ◦ = ǫ ◦ c ◦ F / s = 1 . (101)The fixing of µ ◦ or ǫ ◦ to 1 may also be viewed, equivalently, as a fixing of the impedance of vacuum to1, next to c = 1.We have formulated this analysis by imposing c = 1 and µ ◦ = 1. But we could also have selected,equivalently, any two of the above sets of conditions. With 1 F/m = 1 C /Jm, the supplementary set ofequations in (101) may be used (optionally) as follows : Z ◦ = z ◦ Ω = 1 , to fix 1 Ω = 1 /z ◦ = 1 /µ ◦ c ◦ = ǫ ◦ c ◦ ,ǫ ◦ = ǫ ◦ C / Jm = 1 , to fix 1 C = p J m /ǫ ◦ = 1 / p ǫ ◦ ~ ◦ c ◦ , (102)with the same results as from the fixings of c and/or µ ◦ .However, due to the fixing of the elementary charge as e = e ◦ C, the quantity z ◦ = µ ◦ c ◦ = 1 /ǫ ◦ c ◦ mustbe kept “floating” to adapt to the new value of the ohm, just as for µ ◦ to adapt to the size of the ampere,and ǫ ◦ to that of the coulomb, with z ◦ ∝ η , µ ◦ ∝ η and ǫ ◦ ∝ η − as in (18).2 I. The classical limit
The fundamental laws of nature may now be written in a universal way, no longer referring to theparameters ~ , c, µ ◦ , ǫ ◦ , k, ... , which can all be taken equal to 1, nor to the specific numerical values ~ ◦ , c ◦ , µ ◦ , ǫ ◦ , k ◦ , ... which appear in their expressions when SI units are used. Maxwell’s equations, inparticular, no longer involve µ ◦ and ǫ ◦ , now included within the definitions of the electrical units. TheLorentz transformations relating space and time (or magnetism and electricity) within the frameworkof relativity can also be written without reference to the speed of light c , now equal to 1, through therelation c = c ◦ m/s = 1. The numerical parameter c ◦ gets included within the definition of the metre asobtained from the second, providing the same results as with the usual formalism, thanks to the relation1 metre = (1 /c ◦ ) second.In such a framework with c = ~ = 1 Einstein’s formula E = mc for a free particle at rest furthersimplifies into E = m . For a particle of mass m and momentum p = k we have E = p m + p = p m + k = ω . (103)Now that c = 1 one may enquire about “taking the classical limit” to recover a non-relativistic situation.This can no longer be done by taking a limit “ c → ∞ ”, but is now obtained in the limit of small velocitiesas compared to 1. For small v = dω/dk = dE/dp = p/E , eq. (103) provides back the non-relativisticexpansion of the mechanical energy, as E = m + p / m + ... , without having to consider a limit inwhich the speed of light would become very large. This is similar for quantum effects. We can no longerconsider a limit for which “ ~ → ~ = 1 as the quantum of action (or angular momentum),the classical limit now corresponds to situations involving large values of the action, as compared to theunit quantum. J. The mole and the Avogadro constant
Let us now mention the SI unit of “amount of substance” or mole. In the new SI the mole gets definedby fixing the Avogadro constant to N A = 6 .
022 140 76 × mol − [10]. If n is the amount of substance(in moles) in a sample of X, the number of elementary entities is N = n N A .But the most natural unit for counting “entities” is 1, which fits well in a system of units with ~ = c = µ ◦ = ǫ ◦ = Z ◦ = k = 1. We shall then fix the Avogadro constant at N A = 1, as we did for ~ , c, µ ◦ , ǫ ◦ , Z ◦ and k by fixing their numerical values in SI units to be ~ ◦ , c, µ ◦ , ǫ ◦ , z ◦ and k ◦ , respectively. The Avogadroconstant N A may then be fixed at N A ◦ mol − as in the SI, and, at the same time at N A = 1. This implies N A = 6 .
022 140 76 × | {z } N A ◦ mol − = 1 . (104)This leads us to identify the mole with a pure number, namely the Avogadro number,1 mol = N A ◦ = 6 .
022 140 76 × . (105)The number of elementary entities of a substance X in a sample of n moles is simply N = n N A ◦ .The Avogadro constant N A , normally expressed in mol − , no longer appears as a fundamental constantof nature, but rather as a counting device. Fixing it to the natural unit, 1, as for the other constants,makes the mole appear as a fixed number, like saying that there are twelve eggs in a dozen of eggs.But it is a very large number, the Avogadro number, fixed for consistency with past definitions to N A ◦ = 6 .
022 140 76 × .We may also consider the candela, SI unit of luminous intensity in a given direction, even if we can-not really view it as a fundamental unit. It is now defined and normalized by taking the luminousefficacy of a monochromatic radiation of frequency 540 × Hz (and wavelength λ ≃ . µ ) to be K cd = 683 cd sr W − [4, 10]. Fixing it at 1 as for the other constants, we get K cd = 683 lm W − =683 cd sr W − = 1 . The candela and the lumen appear, for the specific radiation considered, as a certainnumber of watts per steradian or watts, with 1 cd = (1/683) W/sr , 1 lm = (1/683) W.3
XI. CONCLUSIONS
The International System of units is getting redefined, with the joule and kilogram obtained by fixingthe Planck constant at h = 6 .
626 070 15 × − J s . This makes obsolete the international prototype ofthe kilogram stored at BIPM. No longer having to rely on such a single material object for the definitionsof mechanical and electrical units is a huge progress.The coulomb and the other electrical units are also redefined, so that the elementary charge is fixed at e = 1 .
602 176 634 × − C in the new SI. This requires that the vacuum magnetic permeability µ ◦ beslightly adapted, into µ ◦ = 4 π × − η N/A . Fixing the values of h and e in joule · seconds and coulombswill allow for more precise measurements, thanks to quantum electrical metrology based on the Josephsonand quantum Hall effects. Still one may regret that, with µ ◦ no longer exactly fixed but proportional to α , all electrical units, together with the numerical values of the vacuum magnetic permeability (in N/A or H/m), electric permittivity (in F/m) and vacuum impedance (in ohms) become dependent on α .An appealing system should have c = µ ◦ = ǫ ◦ = 1 and ~ = 1, as suggested by relativity and quantummechanics ; but this is usually considered as unpractical as it would naturally lead to units of space andmass ≃ × m and 10 − kg, not very convenient. Still it is possible to reconcile and unite bothsystems, allying the practical interest and convenience of the normal SI units to the advantages andelegance of a symmetric system with c = µ ◦ = ǫ ◦ = ~ = 1 .To this end the fundamental laws of physics, universal, may be expressed without referring to c and ~ , nor to the convention-dependent parameters c ◦ , ~ ◦ , µ ◦ and ǫ ◦ . These now serve to define and resizeappropriately our fundamental units of length, energy and mass, and intensity, all defined from the secondor s − , so that we recover our usual units, metre, joule and kilogram, ampere, suitably normalized.This is done, first, through a rewriting of the laws of electromagnetism by eliminating µ ◦ , and subse-quently ǫ ◦ , from their expressions. µ ◦ becomes a conventional dimensionless normalisation coefficient forthe electrical units, getting included within their expressions. The ampere and the coulomb get given by1 A = p µ ◦ N , p µ ◦ kg m . (106)This is sufficient to derive all electrical units from a choice of µ ◦ . This one is initially fixed to 4 π × − ,multiplied by η , very close to 1, to allow for the necessary adjustement of the ampere and the coulomb,if the elementary charge gets fixed as in (7). This new formulation is well adapted to take in charge thatthe ampere and the coulomb are no longer rigidly tied to the newton, but allowed to slightly “float”proportionally to √ µ ◦ , i.e. to η = p α/α ◦ .By demanding that c and ~ , in addition to being numerically fixed in SI units according to the officialdefinitions, be also equal to 1, we can unite the advantages of both systems through the equations c = c ◦ m / s = 1 ⇒ /
299 792 458) s , ~ = ~ ◦ J s = 1 ⇒ / ~ ◦ ) s − = .
948 252 ... × s − ,µ ◦ = µ ◦ N / A = 1 ⇒ p µ ◦ N = p µ ◦ c ◦ / ~ ◦ s − = 1 .
890 067 ... × s − . (107)One can also, equivalently, define directly the dimensionless ohm and coulomb through one of the condi-tions ( Z ◦ = z ◦ Ω = 1 ⇒ /z ◦ = 1 /µ ◦ c ◦ = ǫ ◦ c ◦ = 1 / . ... ,ǫ ◦ = ǫ ◦ C / N m = 1 ⇒ p N m /ǫ ◦ = p /ǫ ◦ ~ ◦ c ◦ = 1 .
890 067 ... × , (108)allowing for a more symmetric treatment between electricity and magnetism.The coulomb, the ohm, and the weber, related by 1 C × e , the impedance of vacuum Z ◦ , and the flux quantum Φ ◦ expressed as pure numbers : e = 1 .
602 176 634 × − C = √ πα ≃ .
302 822 1208 ,Z ◦ = µ ◦ c ◦ Ω = 376 .
730 313 ...
Ω = 1 , Φ ◦ = h/ e = 2 .
067 833 848 ... × − Wb = π/e ≃ .
374 382 97 . (109)4The impedance of vacuum is 1, as it should with ǫ ◦ = µ ◦ = 1. This is an improvement over the newSI description in which Z ◦ = µ ◦ c ◦ Ω refers to an ohm dependent on µ ◦ and thus now on α , hiding that Z ◦ = 1 is the natural unit of impedance. Fixing the impedance of the vacuum to Z ◦ = 376 . ... Ω = 1also determines the von Klitzing constant, both in ohms and in terms of the fine structure constant,as R K = h/e = 1 / α . Its SI expression in ohms is simply recovered as 376. 730 313 ... Ω (vacuumimpedance) × .
035 9991 ... / R K = h/e = 376 .
730 313 ... Ω / α = 25 812 .
807 459 ...
Ω = 1 / α ≃ .
517 999 57 . (110)The Josephson constant, now also dimensionless, is K J = 1 / Φ ◦ = 2 e/h = 483 597 .
848 416 ...
GHz / V = e/π ≃ . . (111)Inductances and capacitances are naturally expressed in metres owing to their geometric origin. Theycan be converted in ordinary SI units, with inductances measured in henrys proportionally to µ ◦ , andcapacitances in farads proportionally to ǫ ◦ , thanks to the symmetric relations1 m = µ ◦ H = ǫ ◦ F , or 1 s = µ ◦ c ◦ H = ǫ ◦ c ◦ F . (112)These provide back the equalities 1 s = 1 H / Ω = 1 F Ω , and 1 Wb = 1 C × p J / m , are obtained in s − .Extending these ideas to the kelvin and the mole, we get a unified system embedding the new SI withina framework where quantities previously considered as “fundamental constants of nature” return to theirnatural status of being simply 1, i.e. c = ~ = µ ◦ = ǫ ◦ = Z ◦ = k = N A = 1 , (113)and no longer appear within the expressions of the fundamental laws. This is achieved through the set ofequalities, c ◦ m / s = ~ ◦ J s = z ◦ Ω = k ◦ J / K = N A ◦ mol − = 1 . (114)Once the second is chosen as the unit of time, c ◦ fixes the size of the metre, ~ ◦ the joule and kilogram, k ◦ the kelvin, and N A ◦ the mole. Z ◦ = 1 (or µ ◦ or ǫ ◦ = 1) fix the vacuum impedance, magnetic permeabilityand electric permittivity to 1, and the size of the electrical units from µ ◦ (or the exact size of µ ◦ if thecoulomb is obtained from the fixing of e ).We have chosen in (114) to determine the ohm, a dimensionless unit, to be 1 /z ◦ = 1 /µ ◦ c ◦ = 1 / . ... so that the impedance of vacuum is 376 . ... Ω = 1. This also determines R K = 376 . ... Ω / α =25 812 . ... Ω. We could choose as well, to fix the electrical units, one of the equivalent conditions z ◦ H / s = z − ◦ F / s = µ ◦ N / A = ǫ ◦ C / (J s × m / s) = 1 . (115)It is often said that the new SI provides a system of units by fixing the values of fundamental constants ofnature such as the speed of light and the Planck constant. Including it within the proposed system leads totranscend this point of view. We take advantage of the natural set of units provided by quantum mechanicsfor the quantum of action ~ = 1, by relativity for the speed of light c = 1, and by electromagnetism forthe vacuum magnetic permeability µ ◦ = 1, electric permittivity ǫ ◦ = 1, and vacuum impedance Z ◦ =( µ ◦ /ǫ ◦ ) / = 1. We then rescale these natural units, all related to the second or simply equal to 1, to getthe properly normalized units we are used to.Deciding a choice of values for the numerical constants ~ ◦ , c ◦ , and µ ◦ (with related ǫ ◦ and z ◦ ) doesnot fix the Planck constant, the speed of light nor the permeability or the permittivity or the impedanceof vacuum , which all remain identical to 1. But it fixes instead the sizes of the various units (withor without dimensions) we have chosen to use, derived from the second. This is also the case for theBoltzmann constant, providing the usual kelvin, and for the Avogadro constant, turned back into a fixedAvogadro number N A ◦ providing the definition of the mole.5It has been agreed to fix numerically ~ ◦ and c ◦ to define the metre, the joule, and the kilogram,but decided not to fix µ ◦ = 4 π × − (N/A ), to let the coulomb and ampere free to adjust to thechosen value of the elementary charge. This may be practically convenient, but remains conceptuallyquestionable. While the vacuum magnetic permeability, electric permittivity and impedance are all equalto 1, as the speed of light, their numerical values in SI units (N/A or H/m, F/m and Ω, respectively)get now dependent on the elementary charge e , i.e. on future experimental measurements of α . This is asomewhat unpleasant consequence of the fixing of e ◦ in the new SI.This is well managed by embedding the new SI within the proposed system, where the floating cha-racter of electrical units through their dependence on an unfixed µ ◦ proportional to α is made explicitand automatically taken care of. ~ = c = µ ◦ = ǫ ◦ = Z ◦ = k = N A = 1 guarantees that the vacuum magneticpermeability, electric permittivity and impedance all remain constant and identical to 1. The construc-tion combines the advantages of both systems, with a simplified description of the fundamental laws,dimensionless charges, impedances and fluxes, an elementary charge e = √ πα and flux quantum π/e ,the kelvin as a unit of energy, and the mole identified as a very large Avogadro number. 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